Geometry: Translations, Reflections, Rotations, and Dilations

One unit covered in Geometry deals with the concept of translations, reflections, rotations, and dilations. This article will serve to summarize some the major points for students studying these topics.

Prerequisite: The student needs to come into the lesson with some basic understanding of matrices. Given a shape with points on a coordinate plane they need to be able to write those in matrix form. It is very simple actually, you take the x-coordinates and make them the first row of your matrix and take the y-coordinates and make them the second row of your matrix.

Example: A quadrilateral with points (-4,-3), (-1,0), (1,-3), and (-3, -5) would be written as the following matrix:

Dilations: Dilations make an object bigger or smaller. If the dilation is a number bigger than 1, the object will increase in size; if it is less than 1, it will get smaller. Dilations require you multiple each number in the given matrix by the dilation value.

Matrix 2: This matrix looks like the identity but has a negative only in the top 1. This means only the top row will change to their opposite signs but the bottom row will stay the same.

\( \left[
\begin{array}{ c c }
-1 & 0 \\
0 & 1
\end{array} \right]
\)

Matrix 3: Matrix 3 is similar to Matrix 2 but the negative is on the bottom instead of the top. This means the bottom row will change to its opposite sign and the top row stays the same.

\( \left[
\begin{array}{ c c }
1 & 0 \\
0 & -1
\end{array} \right]
\)

Matrix 4: This matrix is a little different from the identity. The 1’s and the 0’s have changed places. When this happens, the whole row changes places. Since both are positive, the numbers keep their original signs.

Matrix 6 – the rows switch place and the bottom row has opposite signs.

Matrix 7 – the rows switch places and both rows also change signs.

Each of these matrices are multiplied times the matrix defined by the shape in the problem Note that the “identity” type matrix always comes first, then the other matrix so that the dimensions match for multiplying.

Apex-Math was started by Dr. Lynne Gregorio. Lynne has been working in the field of mathematics education since 1989. She received a Ph.D. in Mathematics Education from North Carolina State University in 1998. She has taught everyone from Pre-K students to doctoral students. She runs the Apex Learning Center in North Carolina where she tutors students in reading, spelling, writing, and mathematics. She has recently started developing curriculum so that she can reach a broader range of students across the nation. She is married and has 4 children. She is currently homeschooling her oldest son for his senior year of high school.

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